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Analytical solutions of Landau–Lifshitz equation for precessional dynamics
ARTICLE IN PRESS
Physica B 343 (2004) 325–330
Analytical solutions of Landau–Lifshitz equation for precessional dynamics G. Bertottia, I.D. Mayergoyzb, C. Serpicoc,* a Istituto Elettrotecnico Nazionale (IEN) ‘‘Galileo Ferraris’’, Strada delle Cacce 91, I-10135 Torino, Italy Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA c Department of Electrical Engineering, Universita" di Napoli, ‘‘Federico II’’, via Claudio 21, I-80125 Napoli, Italy b
Abstract A rigorous analysis of the precessional magnetization dynamics in uniformly magnetized particles and films is carried out by deriving exact analytic solutions of Landau–Lifshitz equation. The magnetic body is assumed of ellipsoidal shape, the external field is constant in time and applied in the plane normal to the hard axis. The analytic integration of the Landau–Lifshitz equation is based on the explicit knowledge of two integrals of motion for the magnetization dynamics and leads to closed form expressions for the magnetization in terms of Jacobi elliptic functions. r 2003 Elsevier B.V. All rights reserved. PACS: 75.40.Gb; 75.60.Jk; 05.45.a Keywords: Landau–Lifshitz–Gilbert equation; Magnetization dynamics
1. Magnetization dynamics equation The analysis of magnetization dynamics in thin films and magnetic particles is one of the crucial issues in magnetic recording and data storage technologies. In this respect, it has been recognized that the dynamic response of magnetic particles and films to short field pulses is considerably affected by precessional motion of magnetization. The purpose of this paper is to present a rigorous analysis of precessional magnetization motion in uniformly magnetized bodies based on Landau– Lifshitz (LL) equation. This analysis is carried out by deriving analytical solutions of LL equation in the case of constant applied fields and zero *Corresponding author. Tel.: +39-081-7683180. E-mail address: [email protected] (C. Serpico).
damping. The analysis is applicable to all those situations where the magnetic body is subject to short field pulses and the dynamics is so fast that dissipative effects can be neglected during the duration of the pulse [1]. The undamped LL equation is written in dimensionless form as dm ¼ m heff ðmÞ; dt
ð1Þ
where m ¼ M=Ms ; heff ¼ Heff =Ms (normalized effective field), time is measured in units of ðgMs Þ1 ; Ms is the saturation magnetization, g is the absolute value of the gyromagnetic ratio. We assume that the body is of ellipsoidal shape with possible crystal anisotropy contributions along the principal axes of the ellipsoid. These axes will be denoted by x; y; and z; respectively, with associated
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.064
ARTICLE IN PRESS G. Bertotti et al. / Physica B 343 (2004) 325–330
326
unit vectors ex ; ey ; and ez : The magnetization motion takes place on the surface of the unit sphere m2x þ m2y þ m2z ¼ 1:
ð2Þ
The effective field is given by the gradient of the free energy of the system, which, in this case, is of the form gL ðmÞ ¼ 12 Dx m2x þ 12 Dy m2y þ 12 Dz m2z hx mx hy my hz mz ;
ð3Þ
where Dx ; Dy ; Dz are three anisotropy (shape + crystal) coefficients along the three principal directions, and hx ; hy ; hz the components of applied field (in units of Ms ) along x; y; and z: The anisotropy coefficients are ordered as follows, Dx pDy pDz ; thus the easy axis of the particle is along ex : The effective field associated with Eq. (3) is qgL ¼ ðhx Dx mx Þex qm þ ðhy Dy my Þey þ ðhz Dz mz Þez :
heff ¼
ð4Þ
In particular, we will study the case where the field lies in the ðx; yÞ plane (i.e., hz ¼ 0).
this, it can be inferred that there will be two trajectories (or two parts of the same trajectory) with positive and negative mz yielding identical projections onto the ðx; yÞ plane. With this additional piece of information in mind, the phase portrait associated with any generic field value is immediately obtained by drawing the family of all self-similar ellipses of center ðmx ¼ ax ; my ¼ ay Þ and aspect ratio k; and by intersecting that family with the unit circle (for an example of a phase portrait see Fig. 1). When a particular ellipse of the family is tangent to the unit circle, the tangent point represents a fixed point of the dynamics (notice that these fixed points lie on the unit circle and therefore are characterized by mz ¼ 0), since it is a stationary point of the free energy (3) under the constraint (2). More precisely, it can be easily verified that, if the ellipse is tangent to the unit circle from the inside, the equilibrium point is a saddle, while if it is tangent from the outside then the equilibrium is a minima of the free energy (see Fig. 1). The only equilibrium points not lying on the unit circle (and thus with mz a0) are ðmx ¼ ax ; my ¼ ay Þ; existing when a2x þ a2y p1: These equilibria can be viewed as special trajectory with p ¼ 0 and correspond to energy maxima.
2. Trajectories in the ðmx ; my Þ-plane
my The motion described by Eq. (1) yields no energy dissipation, so it will take place along constant-energy trajectories. According to Eqs. (2) and (3), along a trajectory of constant energy g0 ; the projection of the magnetization onto the ðx; yÞ plane satisfies the equation: ðmx ax Þ2 þ k2 ðmy ay Þ2 ¼ p2 ;
ð5Þ
min
L
min
I
L mx
H
where hy ; Dz Dy
hx ; Dz Dx
ay ¼
p2 ¼ a2x þ k2 a2y þ
Dz 2g0 ; Dz Dx
ax ¼
saddle
p=k(1+ay)
k2 ¼
Dz Dy : Dz Dx
max
ð6Þ ð7Þ
Admissible trajectories are obtained by taking that part of the trajectory described by Eq. (5) which lies inside the circle m2x þ m2y p1: Notice that mz appears nowhere in these considerations. From
p=k(1−ay)
Fig. 1. Phase portrait in the ðmx ; my Þ-plane for hx ¼ 0 and hy > 0:
ARTICLE IN PRESS G. Bertotti et al. / Physica B 343 (2004) 325–330
3. Magnetization dependence on time The system trajectories can be expressed in the following parametric form: mx ¼ ax p cos u;
my ¼ ay þ ð p=kÞ sin u;
ð8Þ
where the connection between the parametric variable u and time is to be determined. Substitution into the x-component of Eq. (1) gives du ¼ kðDz Dx Þ mz dt
ð9Þ
and after separation of variables one arrives at the equation du qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðax p cos uÞ2 ðay þ ð p=kÞ sin uÞ2 ¼ kðDz Dx Þ dt:
ð10Þ
Eq. (9) shows once more that the fixed points of the dynamics (i.e., points where u is independent of time) are generally points where mz ¼ 0: The energy maxima with mz a0 previously discussed can violate this rule because in that case p ¼ 0: Let us introduce the variable w ¼ tanðu=2Þ: Then, Eq. (10) can be expressed in terms of w only as dw k pffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðDz Dx Þ dt; P4 ðwÞ 2
ð11Þ
where
327
leads to the expression dw pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 w2 Þðw w Þðwþ wÞ ¼ pk0 ðDz Dx Þ dt; where k02 ¼ 1 k2 ; and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 a2y k ay k w7 ¼ 02 7 0 1 p2 þ 02 : pk pk k
ð13Þ
ð14Þ
In the present case, the roots of the polynomial at the denominator in Eq. (13) are simply expressed since the fourth-order polynomial is factorized into the product of two second-order polynomials. Notice also that under the assumption hy > 0 (ay o0), one has wþ þ w > 0: The behavior of Eq. (13) basically depends on how the four roots of the denominator are ordered. It is therefore useful to discuss how these roots are located on the w-axis for different trajectories. The phase portrait is divided by the saddle separatrices into the following three regions, associated with well-defined energy intervals (see Fig. 1). Region H (high energy) with 0pppkð1 þ ay Þ: In this region w o 1o1owþ and, when p ¼ kð1 þ ay Þ; w ¼ 1 and wþ > 1; while for p ¼ 0 (energy maxima), w - N; wþ - þ N . Region I (intermediate energy) with kð1 þ ay Þpppkð1 ay Þ and 1ow o1ow þ: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Region L with kð1 ay Þppp 1 þ ðk2 a2y =k02 Þ:
ð12Þ
In this region 1ow owþ o1; and, when p ¼ kð1 ay Þ; wþ ¼ 1; w > 1; while for qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ 1 þ ðk2 a2y =k0 2 Þ; wþ ¼ w (energy minima).
is of a fourth-order polynomial. This differential form at the left-hand side of Eq. (11) can be integrated by using elliptic integrals and elliptic functions [2]. Nevertheless, the algebraic manipulations required to transform Eq. (11) into the canonical elliptic integrals are based on the knowledge of the roots of the polynomial P4 ðwÞ which are generally given by rather complicated formulas. In the following we limit ourselves to the case hx ¼ 0; hy > 0; which implies that ax ¼ 0; ay o0: In this case, Eq. (10) can be transformed by using the change of variable w ¼ sin u which
Let us first discuss Regions H and L: Region I will require a slightly different treatment. In order to reduce the left-hand side of Eq. (13) in a form expressible in terms of Jacobi elliptic functions it is useful to carry out a transformation leaving that expression in the same mathematical form (i.e., the differential of the independent variable over the square root of a fourth-order polynomial) but changing the roots of the polynomial. In this respect, we need a transformation leaving the 71 roots of the denominator of Eq. (13) unchanged and bringing the remaining two roots to opposite values. This task can be
P4 ðwÞ ¼ ð1 þ w2 Þ2 ½ax ð1 þ w2 Þ pð1 w2 Þ2 ½ay ð1 þ w2 Þ þ ð2p=kÞw2
ARTICLE IN PRESS G. Bertotti et al. / Physica B 343 (2004) 325–330
328
. obtained by the following Mobius transformation with fixed points at 71 [3] x¼
wq ; 1qw
w¼
xþq ; 1þqx
ð15Þ
where the parameter q has to be properly set to get symmetric roots in the x axis. In fact, given x7 ¼ ðw7 qÞ=ð1 q w7 Þ; we want xþ þ x ¼ 0 and this can be obtained by choosing the parameter q fulfilling the equation q2 2sq þ 1 ¼ 0; where s ¼ ð1 þ wþ w Þ=ðwþ þ w Þ: Thus, there are two values of p q ffiffiffiffiffiffiffiffiffiffiffiffi which ffi fulfill the above condition q ¼ s 7 s2 1 which are real only when s2 1X0: It turns out that s2 1 ¼ ðw2þ 1Þðw2 1Þ=ðwþ þ w Þ2 and therefore s2 1X0 when the w7 roots are both outside or both inside the interval ð1; 1Þ which is precisely what occurs in Regions H and L; . respectively. The Mobius transformation (15) has a determinant equal to 1 q2 and, from the property of this class of transformation, one can derive that 1 q2 must be positive if we want that the order of the roots on the w-axis is preserved on the x-axis (basically 1 q2 > 0 guarantees that the transformation (15) is strictly increasing in the interval of interest). This further requirement can be fulfilled by appropriately choosing one of the two admissible values for q: In particular, we notice that in the region H; so0 and p 1 ffiffiffiffiffiffiffiffiffiffiffiffi q2 > ffi 0 is fulfilled by choosing q ¼ qH ¼ s þ s2 1; while in the region L; s > 0 andpthe appropriate value of q is q ¼ qL ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffi s s 2 1: Let us now transform Eq. (13) according to the new variable x assuming that we are in the region H (analogous derivation can be carried out in the region L). It can be derived that dx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ OH dt; 2 2 x2 Þ ð1 x Þð1 kH
ð16Þ
where 1 1 qH wþ 1 qH w ¼ ¼ ; xþ wþ qH w qH sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2H OH ¼ pk0 ðDz Dx Þ : 2 q2 kH H
kH ¼
The latter ðqH w Þðwþ qH Þ=ð1 q2H ÞX0: quantity is positive since w oqH owþ for the order-preserving property of Eq. (15). By integrating Eq. (16) one obtains a solution for x in terms of Jacobi elliptic functions [2]: x ¼ snðOH t; kH Þ:
ð18Þ
Knowledge of x permits one to calculate all the magnetization components through the following expressions: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi cnH mx ¼ 8p 1 q2H ; 1 þ qH snH p qH þ snH ; ð19Þ my ¼ ay þ k 1 þ qH snH pk0 mz ¼ 7
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2H 1 q2H dnH 2 2 1 þ qH snH k kH qH
ð20Þ
where snH ; cnH ; and dnH denote the corresponding elliptic functions of argument ðOH t; kH Þ: The presence of signs þ and is related to the symmetry of the problem and to the fact that for a given energy two different solutions may exist depending on the initial conditions. The analytical solution for magnetization motion in the Region L can be derived with a very similar line of reasoning. The only basic change is that now the value qL of q must be chosen instead of qH : Moreover, if we apply the above solution to Region L we have that the modulus of the elliptic integral, k*L ¼ 1=xþ is now larger that 1: In this respect, one can define a new modulus kL2 ¼ 1=k*2L and express the Jacobi elliptic functions in terms of the new modulus by applying well-known identities. The analytical solution can be expressed as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi dnL mx ¼ 8p 1 q2L ; 1 þ qL kL snL p qL þ kL snL my ¼ ay þ ; ð21Þ k 1 þ qL kL snL pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2L * mz ¼ 7 kL k sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pk0
ð17Þ
In the above equation one has 0okH o1 2 (since xþ > 1) and ð1 q2H Þ=ðkH q2H Þ ¼
1 q2L kL cnL ; k*2L q2L 1 þ qL kL snL
ð22Þ
ARTICLE IN PRESS G. Bertotti et al. / Physica B 343 (2004) 325–330
where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2L 0 OL ¼ pk ðDz Dx Þk*L k*2L q2L
ð23Þ
and snL ; cnL ; and dnL denote the corresponding elliptic functions of argument ðOL t; kL Þ: Before passing to the analysis of the Region I; it is appropriate here to provide a brief analysis of a relevant special case: the case of the free magnetization precession, i.e. when no field is applied. In this case ax ¼ ay ¼ 0 and the interval of value that defines Region I; kð1 þ ay Þpppkð1 ay Þ; becomes empty. This means that in the case of free precession there are only the two regions H and L: The analytical solutions simplify considerably. In fact, for ay ¼ 0; we have wþ ¼ w ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 p2 =ð pk0 Þ and both qL and qH vanish. This in turn implies that the transformation (15) reduces to the identity and thus xþ ¼ wþ ; kH ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi k*L ¼ 1=wþ ¼ pk0 = k 1 p2 : The analytical solutions assume the following simple form. In the Region H; 0pppk; p mx ¼ 8p cnðOH t; kH Þ; my ¼ snðOH t; kH Þ; k pffiffiffiffiffiffiffiffiffiffiffiffiffi ð24Þ mz ¼ 7 1 p2 dnðOH t; kH Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffi where OH ¼ k 1 p2 ðDz Dx Þ: In the Region L; kppp1; mx ¼ 8p dnðOL t; kL Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p2 my ¼ snðOL t; kL Þ; 2 k0ffiffiffiffiffiffiffiffiffiffiffiffiffi p ð25Þ mz ¼ 7 1 p2 cnðOH t; kH Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffi where OL ¼ kH OH ¼ kH k 1 p2 ðDz Dx Þ; kL ¼ 1=kH : Let us note that the LL equation for the free precession is formally identical to the Euler equation for a rigid body moving about a fixed point, which is a well-known example of integrable dynamical system. Accordingly, one can verify that the analytical solutions are coincident (see Ref. [4]). Now we pass to the analysis of Region I: The reduction of the left-hand side of Eq. (16) into a canonical elliptic form can be achieved in two steps. First the ordering 1ow o1owþ is o 1o1ow by a transfortransformed into w þ mation leaving 1 unchanged and bringing
329
w to 1: This intermediate transformation is 1w w ¼ 1 2 ; w ¼ 1 þ c ðw 1Þ; ð26Þ 1 w where c ¼ ð1 w Þ=2: In terms of w ; Eq. (13) becomes dw pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 w2 Þðw w Þðwþ w Þ ¼ cpk0 ðDz Dx Þ dt;
ð27Þ
where ¼1 w
4 ; 1 w
1 wþ wþ ¼ 1 2 : 1 w
ð28Þ
o 1o1ow ; can be solved by Eq. (27), with w þ the method used for Eq. (13) in the case of region H: By introducing everywhere starred quantities in the place of the original ones, one can immediately write the solution as x ¼ snðOI t; kI Þ;
ð29Þ
where 1 qI wþ 1 1 qI w kI ¼ ¼ ¼ ; w þ qI w qI xþ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2I 0 OI ¼ cpk ðDz Dx Þ ; kI2 q2I pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qI ¼ s þ s2 1; Þ=ðw þ w Þ: s ¼ ð1 þ wþ w þ
ð30Þ
ð31Þ
The connection between x and the magnetization components is more complicated than before, because of the intermediate transformation from w to w : After certain algebraic manipulations and with an appropriate change of the time origin the following analytical expressions are obtained: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ qI mx ¼ cpð1 qI Þ k I qI pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdnI kI cnI Þð1 þ dnI Þ cnðOI t=2; kI Þ; dnI qI cnI ð32Þ
ARTICLE IN PRESS 330
G. Bertotti et al. / Physica B 343 (2004) 325–330
p cpð1 qI Þ dnI þ cnI m y ¼ ay þ ; k k dnI qI cnI sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cpk0 ð1 þ qI Þ 1 qI mz ¼ k k I þ qI pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdnI þ kI cnI Þð1 þ dnI Þ dnI qI cnI 0 k snðOI t=2; kI Þ ; I dnðOI t=2; kI Þ
ð33Þ
Acknowledgements This work is partially supported by US Department of Energy and by Italian MIUR-FIRB (contract No. RBAU01B2T8).
References ð34Þ
where kI02 ¼ 1 kI2 ; and snI ; cnI ; and dnI denote the corresponding elliptic functions of argument ðOI t; kI Þ:
[1] C. Serpico, I.D. Mayergoyz, G. Bertotti, J. Appl. Phys. 93, 6909. [2] H. Hancock, Elliptic Integrals, Dover, New York, 1958. [3] H. Schwerdtfeger, Geometry of Complex Number, Dover, New York, 1979. [4] L.D. Landau, E.M. Lifshitz, Mechanics, Course of Theoretical Physics, Pergamon Press, New York, 1960.
Physica B 343 (2004) 325–330
Analytical solutions of Landau–Lifshitz equation for precessional dynamics G. Bertottia, I.D. Mayergoyzb, C. Serpicoc,* a Istituto Elettrotecnico Nazionale (IEN) ‘‘Galileo Ferraris’’, Strada delle Cacce 91, I-10135 Torino, Italy Department of Electrical and Computer Engineering, University of Maryland, College Park, MD 20742, USA c Department of Electrical Engineering, Universita" di Napoli, ‘‘Federico II’’, via Claudio 21, I-80125 Napoli, Italy b
Abstract A rigorous analysis of the precessional magnetization dynamics in uniformly magnetized particles and films is carried out by deriving exact analytic solutions of Landau–Lifshitz equation. The magnetic body is assumed of ellipsoidal shape, the external field is constant in time and applied in the plane normal to the hard axis. The analytic integration of the Landau–Lifshitz equation is based on the explicit knowledge of two integrals of motion for the magnetization dynamics and leads to closed form expressions for the magnetization in terms of Jacobi elliptic functions. r 2003 Elsevier B.V. All rights reserved. PACS: 75.40.Gb; 75.60.Jk; 05.45.a Keywords: Landau–Lifshitz–Gilbert equation; Magnetization dynamics
1. Magnetization dynamics equation The analysis of magnetization dynamics in thin films and magnetic particles is one of the crucial issues in magnetic recording and data storage technologies. In this respect, it has been recognized that the dynamic response of magnetic particles and films to short field pulses is considerably affected by precessional motion of magnetization. The purpose of this paper is to present a rigorous analysis of precessional magnetization motion in uniformly magnetized bodies based on Landau– Lifshitz (LL) equation. This analysis is carried out by deriving analytical solutions of LL equation in the case of constant applied fields and zero *Corresponding author. Tel.: +39-081-7683180. E-mail address: [email protected] (C. Serpico).
damping. The analysis is applicable to all those situations where the magnetic body is subject to short field pulses and the dynamics is so fast that dissipative effects can be neglected during the duration of the pulse [1]. The undamped LL equation is written in dimensionless form as dm ¼ m heff ðmÞ; dt
ð1Þ
where m ¼ M=Ms ; heff ¼ Heff =Ms (normalized effective field), time is measured in units of ðgMs Þ1 ; Ms is the saturation magnetization, g is the absolute value of the gyromagnetic ratio. We assume that the body is of ellipsoidal shape with possible crystal anisotropy contributions along the principal axes of the ellipsoid. These axes will be denoted by x; y; and z; respectively, with associated
0921-4526/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2003.08.064
ARTICLE IN PRESS G. Bertotti et al. / Physica B 343 (2004) 325–330
326
unit vectors ex ; ey ; and ez : The magnetization motion takes place on the surface of the unit sphere m2x þ m2y þ m2z ¼ 1:
ð2Þ
The effective field is given by the gradient of the free energy of the system, which, in this case, is of the form gL ðmÞ ¼ 12 Dx m2x þ 12 Dy m2y þ 12 Dz m2z hx mx hy my hz mz ;
ð3Þ
where Dx ; Dy ; Dz are three anisotropy (shape + crystal) coefficients along the three principal directions, and hx ; hy ; hz the components of applied field (in units of Ms ) along x; y; and z: The anisotropy coefficients are ordered as follows, Dx pDy pDz ; thus the easy axis of the particle is along ex : The effective field associated with Eq. (3) is qgL ¼ ðhx Dx mx Þex qm þ ðhy Dy my Þey þ ðhz Dz mz Þez :
heff ¼
ð4Þ
In particular, we will study the case where the field lies in the ðx; yÞ plane (i.e., hz ¼ 0).
this, it can be inferred that there will be two trajectories (or two parts of the same trajectory) with positive and negative mz yielding identical projections onto the ðx; yÞ plane. With this additional piece of information in mind, the phase portrait associated with any generic field value is immediately obtained by drawing the family of all self-similar ellipses of center ðmx ¼ ax ; my ¼ ay Þ and aspect ratio k; and by intersecting that family with the unit circle (for an example of a phase portrait see Fig. 1). When a particular ellipse of the family is tangent to the unit circle, the tangent point represents a fixed point of the dynamics (notice that these fixed points lie on the unit circle and therefore are characterized by mz ¼ 0), since it is a stationary point of the free energy (3) under the constraint (2). More precisely, it can be easily verified that, if the ellipse is tangent to the unit circle from the inside, the equilibrium point is a saddle, while if it is tangent from the outside then the equilibrium is a minima of the free energy (see Fig. 1). The only equilibrium points not lying on the unit circle (and thus with mz a0) are ðmx ¼ ax ; my ¼ ay Þ; existing when a2x þ a2y p1: These equilibria can be viewed as special trajectory with p ¼ 0 and correspond to energy maxima.
2. Trajectories in the ðmx ; my Þ-plane
my The motion described by Eq. (1) yields no energy dissipation, so it will take place along constant-energy trajectories. According to Eqs. (2) and (3), along a trajectory of constant energy g0 ; the projection of the magnetization onto the ðx; yÞ plane satisfies the equation: ðmx ax Þ2 þ k2 ðmy ay Þ2 ¼ p2 ;
ð5Þ
min
L
min
I
L mx
H
where hy ; Dz Dy
hx ; Dz Dx
ay ¼
p2 ¼ a2x þ k2 a2y þ
Dz 2g0 ; Dz Dx
ax ¼
saddle
p=k(1+ay)
k2 ¼
Dz Dy : Dz Dx
max
ð6Þ ð7Þ
Admissible trajectories are obtained by taking that part of the trajectory described by Eq. (5) which lies inside the circle m2x þ m2y p1: Notice that mz appears nowhere in these considerations. From
p=k(1−ay)
Fig. 1. Phase portrait in the ðmx ; my Þ-plane for hx ¼ 0 and hy > 0:
ARTICLE IN PRESS G. Bertotti et al. / Physica B 343 (2004) 325–330
3. Magnetization dependence on time The system trajectories can be expressed in the following parametric form: mx ¼ ax p cos u;
my ¼ ay þ ð p=kÞ sin u;
ð8Þ
where the connection between the parametric variable u and time is to be determined. Substitution into the x-component of Eq. (1) gives du ¼ kðDz Dx Þ mz dt
ð9Þ
and after separation of variables one arrives at the equation du qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ðax p cos uÞ2 ðay þ ð p=kÞ sin uÞ2 ¼ kðDz Dx Þ dt:
ð10Þ
Eq. (9) shows once more that the fixed points of the dynamics (i.e., points where u is independent of time) are generally points where mz ¼ 0: The energy maxima with mz a0 previously discussed can violate this rule because in that case p ¼ 0: Let us introduce the variable w ¼ tanðu=2Þ: Then, Eq. (10) can be expressed in terms of w only as dw k pffiffiffiffiffiffiffiffiffiffiffiffi ¼ ðDz Dx Þ dt; P4 ðwÞ 2
ð11Þ
where
327
leads to the expression dw pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 w2 Þðw w Þðwþ wÞ ¼ pk0 ðDz Dx Þ dt; where k02 ¼ 1 k2 ; and sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k2 a2y k ay k w7 ¼ 02 7 0 1 p2 þ 02 : pk pk k
ð13Þ
ð14Þ
In the present case, the roots of the polynomial at the denominator in Eq. (13) are simply expressed since the fourth-order polynomial is factorized into the product of two second-order polynomials. Notice also that under the assumption hy > 0 (ay o0), one has wþ þ w > 0: The behavior of Eq. (13) basically depends on how the four roots of the denominator are ordered. It is therefore useful to discuss how these roots are located on the w-axis for different trajectories. The phase portrait is divided by the saddle separatrices into the following three regions, associated with well-defined energy intervals (see Fig. 1). Region H (high energy) with 0pppkð1 þ ay Þ: In this region w o 1o1owþ and, when p ¼ kð1 þ ay Þ; w ¼ 1 and wþ > 1; while for p ¼ 0 (energy maxima), w - N; wþ - þ N . Region I (intermediate energy) with kð1 þ ay Þpppkð1 ay Þ and 1ow o1ow þ: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Region L with kð1 ay Þppp 1 þ ðk2 a2y =k02 Þ:
ð12Þ
In this region 1ow owþ o1; and, when p ¼ kð1 ay Þ; wþ ¼ 1; w > 1; while for qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ¼ 1 þ ðk2 a2y =k0 2 Þ; wþ ¼ w (energy minima).
is of a fourth-order polynomial. This differential form at the left-hand side of Eq. (11) can be integrated by using elliptic integrals and elliptic functions [2]. Nevertheless, the algebraic manipulations required to transform Eq. (11) into the canonical elliptic integrals are based on the knowledge of the roots of the polynomial P4 ðwÞ which are generally given by rather complicated formulas. In the following we limit ourselves to the case hx ¼ 0; hy > 0; which implies that ax ¼ 0; ay o0: In this case, Eq. (10) can be transformed by using the change of variable w ¼ sin u which
Let us first discuss Regions H and L: Region I will require a slightly different treatment. In order to reduce the left-hand side of Eq. (13) in a form expressible in terms of Jacobi elliptic functions it is useful to carry out a transformation leaving that expression in the same mathematical form (i.e., the differential of the independent variable over the square root of a fourth-order polynomial) but changing the roots of the polynomial. In this respect, we need a transformation leaving the 71 roots of the denominator of Eq. (13) unchanged and bringing the remaining two roots to opposite values. This task can be
P4 ðwÞ ¼ ð1 þ w2 Þ2 ½ax ð1 þ w2 Þ pð1 w2 Þ2 ½ay ð1 þ w2 Þ þ ð2p=kÞw2
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328
. obtained by the following Mobius transformation with fixed points at 71 [3] x¼
wq ; 1qw
w¼
xþq ; 1þqx
ð15Þ
where the parameter q has to be properly set to get symmetric roots in the x axis. In fact, given x7 ¼ ðw7 qÞ=ð1 q w7 Þ; we want xþ þ x ¼ 0 and this can be obtained by choosing the parameter q fulfilling the equation q2 2sq þ 1 ¼ 0; where s ¼ ð1 þ wþ w Þ=ðwþ þ w Þ: Thus, there are two values of p q ffiffiffiffiffiffiffiffiffiffiffiffi which ffi fulfill the above condition q ¼ s 7 s2 1 which are real only when s2 1X0: It turns out that s2 1 ¼ ðw2þ 1Þðw2 1Þ=ðwþ þ w Þ2 and therefore s2 1X0 when the w7 roots are both outside or both inside the interval ð1; 1Þ which is precisely what occurs in Regions H and L; . respectively. The Mobius transformation (15) has a determinant equal to 1 q2 and, from the property of this class of transformation, one can derive that 1 q2 must be positive if we want that the order of the roots on the w-axis is preserved on the x-axis (basically 1 q2 > 0 guarantees that the transformation (15) is strictly increasing in the interval of interest). This further requirement can be fulfilled by appropriately choosing one of the two admissible values for q: In particular, we notice that in the region H; so0 and p 1 ffiffiffiffiffiffiffiffiffiffiffiffi q2 > ffi 0 is fulfilled by choosing q ¼ qH ¼ s þ s2 1; while in the region L; s > 0 andpthe appropriate value of q is q ¼ qL ¼ ffiffiffiffiffiffiffiffiffiffiffiffi ffi s s 2 1: Let us now transform Eq. (13) according to the new variable x assuming that we are in the region H (analogous derivation can be carried out in the region L). It can be derived that dx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ OH dt; 2 2 x2 Þ ð1 x Þð1 kH
ð16Þ
where 1 1 qH wþ 1 qH w ¼ ¼ ; xþ wþ qH w qH sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2H OH ¼ pk0 ðDz Dx Þ : 2 q2 kH H
kH ¼
The latter ðqH w Þðwþ qH Þ=ð1 q2H ÞX0: quantity is positive since w oqH owþ for the order-preserving property of Eq. (15). By integrating Eq. (16) one obtains a solution for x in terms of Jacobi elliptic functions [2]: x ¼ snðOH t; kH Þ:
ð18Þ
Knowledge of x permits one to calculate all the magnetization components through the following expressions: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi cnH mx ¼ 8p 1 q2H ; 1 þ qH snH p qH þ snH ; ð19Þ my ¼ ay þ k 1 þ qH snH pk0 mz ¼ 7
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2H 1 q2H dnH 2 2 1 þ qH snH k kH qH
ð20Þ
where snH ; cnH ; and dnH denote the corresponding elliptic functions of argument ðOH t; kH Þ: The presence of signs þ and is related to the symmetry of the problem and to the fact that for a given energy two different solutions may exist depending on the initial conditions. The analytical solution for magnetization motion in the Region L can be derived with a very similar line of reasoning. The only basic change is that now the value qL of q must be chosen instead of qH : Moreover, if we apply the above solution to Region L we have that the modulus of the elliptic integral, k*L ¼ 1=xþ is now larger that 1: In this respect, one can define a new modulus kL2 ¼ 1=k*2L and express the Jacobi elliptic functions in terms of the new modulus by applying well-known identities. The analytical solution can be expressed as follows: qffiffiffiffiffiffiffiffiffiffiffiffiffiffi dnL mx ¼ 8p 1 q2L ; 1 þ qL kL snL p qL þ kL snL my ¼ ay þ ; ð21Þ k 1 þ qL kL snL pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2L * mz ¼ 7 kL k sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pk0
ð17Þ
In the above equation one has 0okH o1 2 (since xþ > 1) and ð1 q2H Þ=ðkH q2H Þ ¼
1 q2L kL cnL ; k*2L q2L 1 þ qL kL snL
ð22Þ
ARTICLE IN PRESS G. Bertotti et al. / Physica B 343 (2004) 325–330
where
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2L 0 OL ¼ pk ðDz Dx Þk*L k*2L q2L
ð23Þ
and snL ; cnL ; and dnL denote the corresponding elliptic functions of argument ðOL t; kL Þ: Before passing to the analysis of the Region I; it is appropriate here to provide a brief analysis of a relevant special case: the case of the free magnetization precession, i.e. when no field is applied. In this case ax ¼ ay ¼ 0 and the interval of value that defines Region I; kð1 þ ay Þpppkð1 ay Þ; becomes empty. This means that in the case of free precession there are only the two regions H and L: The analytical solutions simplify considerably. In fact, for ay ¼ 0; we have wþ ¼ w ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 p2 =ð pk0 Þ and both qL and qH vanish. This in turn implies that the transformation (15) reduces to the identity and thus xþ ¼ wþ ; kH ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi k*L ¼ 1=wþ ¼ pk0 = k 1 p2 : The analytical solutions assume the following simple form. In the Region H; 0pppk; p mx ¼ 8p cnðOH t; kH Þ; my ¼ snðOH t; kH Þ; k pffiffiffiffiffiffiffiffiffiffiffiffiffi ð24Þ mz ¼ 7 1 p2 dnðOH t; kH Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffi where OH ¼ k 1 p2 ðDz Dx Þ: In the Region L; kppp1; mx ¼ 8p dnðOL t; kL Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p2 my ¼ snðOL t; kL Þ; 2 k0ffiffiffiffiffiffiffiffiffiffiffiffiffi p ð25Þ mz ¼ 7 1 p2 cnðOH t; kH Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffi where OL ¼ kH OH ¼ kH k 1 p2 ðDz Dx Þ; kL ¼ 1=kH : Let us note that the LL equation for the free precession is formally identical to the Euler equation for a rigid body moving about a fixed point, which is a well-known example of integrable dynamical system. Accordingly, one can verify that the analytical solutions are coincident (see Ref. [4]). Now we pass to the analysis of Region I: The reduction of the left-hand side of Eq. (16) into a canonical elliptic form can be achieved in two steps. First the ordering 1ow o1owþ is o 1o1ow by a transfortransformed into w þ mation leaving 1 unchanged and bringing
329
w to 1: This intermediate transformation is 1w w ¼ 1 2 ; w ¼ 1 þ c ðw 1Þ; ð26Þ 1 w where c ¼ ð1 w Þ=2: In terms of w ; Eq. (13) becomes dw pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 w2 Þðw w Þðwþ w Þ ¼ cpk0 ðDz Dx Þ dt;
ð27Þ
where ¼1 w
4 ; 1 w
1 wþ wþ ¼ 1 2 : 1 w
ð28Þ
o 1o1ow ; can be solved by Eq. (27), with w þ the method used for Eq. (13) in the case of region H: By introducing everywhere starred quantities in the place of the original ones, one can immediately write the solution as x ¼ snðOI t; kI Þ;
ð29Þ
where 1 qI wþ 1 1 qI w kI ¼ ¼ ¼ ; w þ qI w qI xþ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 q2I 0 OI ¼ cpk ðDz Dx Þ ; kI2 q2I pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qI ¼ s þ s2 1; Þ=ðw þ w Þ: s ¼ ð1 þ wþ w þ
ð30Þ
ð31Þ
The connection between x and the magnetization components is more complicated than before, because of the intermediate transformation from w to w : After certain algebraic manipulations and with an appropriate change of the time origin the following analytical expressions are obtained: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ qI mx ¼ cpð1 qI Þ k I qI pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdnI kI cnI Þð1 þ dnI Þ cnðOI t=2; kI Þ; dnI qI cnI ð32Þ
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p cpð1 qI Þ dnI þ cnI m y ¼ ay þ ; k k dnI qI cnI sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cpk0 ð1 þ qI Þ 1 qI mz ¼ k k I þ qI pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðdnI þ kI cnI Þð1 þ dnI Þ dnI qI cnI 0 k snðOI t=2; kI Þ ; I dnðOI t=2; kI Þ
ð33Þ
Acknowledgements This work is partially supported by US Department of Energy and by Italian MIUR-FIRB (contract No. RBAU01B2T8).
References ð34Þ
where kI02 ¼ 1 kI2 ; and snI ; cnI ; and dnI denote the corresponding elliptic functions of argument ðOI t; kI Þ:
[1] C. Serpico, I.D. Mayergoyz, G. Bertotti, J. Appl. Phys. 93, 6909. [2] H. Hancock, Elliptic Integrals, Dover, New York, 1958. [3] H. Schwerdtfeger, Geometry of Complex Number, Dover, New York, 1979. [4] L.D. Landau, E.M. Lifshitz, Mechanics, Course of Theoretical Physics, Pergamon Press, New York, 1960.